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Theorem map0b 7938
 Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
map0b (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅)

Proof of Theorem map0b
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elmapi 7921 . . . 4 (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝑓:𝐴⟶∅)
2 fdm 6089 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴)
3 frn 6091 . . . . . . 7 (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅)
4 ss0 4007 . . . . . . 7 (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅)
53, 4syl 17 . . . . . 6 (𝑓:𝐴⟶∅ → ran 𝑓 = ∅)
6 dm0rn0 5374 . . . . . 6 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
75, 6sylibr 224 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = ∅)
82, 7eqtr3d 2687 . . . 4 (𝑓:𝐴⟶∅ → 𝐴 = ∅)
91, 8syl 17 . . 3 (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝐴 = ∅)
109necon3ai 2848 . 2 (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑𝑚 𝐴))
1110eq0rdv 4012 1 (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030   ≠ wne 2823   ⊆ wss 3607  ∅c0 3948  dom cdm 5143  ran crn 5144  ⟶wf 5922  (class class class)co 6690   ↑𝑚 cmap 7899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901 This theorem is referenced by:  map0g  7939  mapdom2  8172  ply1plusgfvi  19660
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